RUSSELL HANSON, DAVID BOHM AND OTHERS ON THE SEMANTICS OF DISCOVERY
BOOK VII - Page 2
Bohm's Hidden-Variable Interpretation of Quantum Theory
Consider next a brief overview of Bohm’s hidden-variable interpretation, his means for implementing his three-point agenda for future microphysics. Bohm’s hidden-variable interpretation is the Schrödinger wave equation plus trajectories for individual particles, as in Newton’s second law of motion, thus rendering both the Newtonian wave and particle as real and completely causal. Contrary to the Copenhagen school Bohm says that measurement does not realize the particle, and there is no collapse of the wave into a particle. Bohm postulates that there exists a subquantum order of magnitude containing hidden phenomena and that the statistical character of the current quantum theory originates in random fluctuations of new kinds of entities existing at this lower subquantum order. Thus Heisenberg’s indeterminacy principle and his particular statistical treatment of it pertain only to phenomena at the quantum-mechanical level. Like Einstein, Bohm believed that indeterminacy is a measurement problem like the measurement problems found in Newtonian mechanics, and that by broadening the context of physical theory to include a subquantum order, it will become possible to diminish indeterminacy below the limits set by Heisenberg’s indeterminacy principle.
Bohm states that subquantum processes may be detectable in the domain of very high energies and very small distances, even though at lesser energies and greater distances the high degree of approximation permitted by the laws of the quantum level means that the entities at the subquantum level cannot be playing a very significant rôle in quantum-level events. He postulates that associated with each electron there is a particle that has a precisely definable and continuously varying values of position and momentum, and that are so small that at the quantum-mechanical level they can be approximated as a mathematical point, just as in the earliest forms of atomic theory the atom was so described. He also postulates that associated with the particle there is a quantum-level wave that oscillates in a real subquantum field, and which satisfies the Schrödinger wave function. In his later works he also refers to the subquantum field as the quantum field. In summary Bohm says he regards the quantum-mechanical system to be a synthesis of a precisely definable particle and a precisely definable subquantum field which exerts a force or potential on the particle.
Bohm uses linguistic figures of speech, which he calls analogies, which are not merely illustrative of fully formed thoughts, but have had a self-consciously formative rôle in his thinking. In his Causality and Chance he uses an analogy with Brownian movements of particles in a gravitational field, and illustrates what Heisenberg’s indeterminacy principle would mean in terms of a subquantum-mechanical field. In the case of Brownian motion a smoke particle is subject to random fluctuations originating in collisions with the atoms that exist at a lower order of magnitude than the smoke particle. As a result of these random collisions the motion of the smoke particle cannot be completely determined by the position and velocity of the particle at the level of the Brownian motion itself. Bohm cites a 1933 paper by the German physicist, R. Furth, who showed that the lack of determination in Brownian motion is not only qualitatively analogous to that obtained in the quantum theory, but is also quantitatively analogous to the mathematical form of the indeterminacy relations. Thus, for a short time interval with random fluctuations of a given magnitude in the mean position and a given magnitude in the mean momentum, the magnitudes satisfy a relationship involving a constant that depends on the state of the gas, and the relationship is mathematically analogous to Heisenberg’s indeterminacy relation involving Planck’s constant.
The quantum-level force produces a tendency to pull the particle into regions where the subquantum-level field has its strongest intensity, as described by Born’s probability distribution. But this tendency is also resisted by random motions analogous to Brownian motions, which originate at the subquantum level. The origin of these motions is not important; it is sufficient they have the property such that the average of their motions satisfies the Schrödinger wave equation, and that they are communicated to the particle. The net effect of the quantum-level force and the subquantum-level random motions in the subquantum field is a mean distribution in a statistical ensemble of particles, which favors the regions where the quantum-level force field is most intense, but which still leaves some chance for a typical particle to spend some time in the regions where the field is relatively weak. This result is analogous to the classical Brownian motion of a particle in a gravitational field, where the random motion which tends to carry the particle into all parts of the container, is opposed by the gravitational field, which tends to pull it towards the bottom of the container.
Using these concepts Bohm proposes his alternative explanation of the two-slit experiment. When the particle passes through a slit, it follows an irregular path, because subquantum random motions affect it. After a large number of particles have passed through the slit system with both slits open, a pattern forms with particles accumulating on a screen where the subquantum field intensity is greatest due to the effects of the quantum force, as described by Born’s probability distribution. The pattern is different if only one slit is open, than if both are open. Closing one of the two slits influences the particles that pass through the open slit, because it influences the quantum-level force affecting the particle as it moves between the slit system and the screen. Thus Bohm claims that the hidden-variable interpretation can explain how the appearance of the wave-particle duality originates, while the Copenhagen interpretation requires acceptance without further discussion of the fact that electrons enter the slit system and appear at the screen with an interference pattern.
In Causality and Chance Bohm also comments on Heisenberg’s gamma-ray microscope thought experiment. He maintains that Heisenberg’s indeterminacy principle should not be regarded as expressing the impossibility of making measurements of unlimited precision. Rather it should be regarded as expressing the incomplete degree of self-determination characteristic only of entities that can be defined in the quantum-mechanical level. The subquantum-mechanical processes involving very small intervals of time and space will not be subject to the same limitations as those of the quantum-mechanical processes, and the unpredictable and uncontrollable disturbances caused by a measurement apparatus at the quantum level can either be eliminated or be controlled and corrected. Thus when the physicist measures processes at the quantum-mechanical level, the process of measurement will have the same limits on its degree of self-determination as every other process at this level. But if the microphysical theory is generalized to include the subquantum order of magnitude, then the problem of measurement attributed to the indeterminacy principle should be regarded not as an inherent limitation on the precision with which it is possible to conceive the simultaneous definition of position and momentum, but rather as merely a practical limitation, because measurement precision in violation of the indeterminacy relations is conceivable.
Bohm also makes another analogy with Brownian motion comparing quantum phenomena with Brownian motion by describing the wave and particle as entities that interact in a way that is essential to their modes of being. He says that this seems plausible, because the fact that wave and particle are never found separately suggests that they are both different aspects of the some fundamentally new kind of entity that is likely to be quite different from a simple wave or particle. Thus if Brownian motion were viewed not as the motion of particles, but as the motion of a very fine droplet of mist, then the indeterminacy of the droplets in the vapor at its critical temperature, where the distinction between liquid and gaseous states disappears, is a fluctuation in which the droplets are always forming and disappearing.
This is an indeterminacy in the very existence of the droplets. Analogously at the quantum level it may be found that the very mode of existence of the electron is indeterminate, because its characteristic wave-particle duality suggests that the particle showing this critical opalescence is the relevant concept of particle. It is unclear whether or not Bohm is attempting at this stage of his thinking in terms of hidden variables to use this alternative Brownian analogy to reconcile his original potentiality idea when he accepted the Copenhagen interpretation with a newer hidden-variables idea, because later in Undivided Universe he portrays potentiality as the presence of information in the quantum wave, which is inactive except when the particle uses it as a guidance condition for its movement.
Bohm says that because the subquantum level is inadmissible in the Copenhagen interpretation, one is restricted to making blind mathematical manipulations with the hope that somehow one of these manipulations will lead to a new and superior theory. But he maintains that if the subquantum level is admitted, where there are processes of very high energy and very high frequency faster than the processes taking place at the quantum-mechanical level, then the details of the lower level would become significant, and the current formulation of the quantum theory would break down.
The creation of a particle such as a meson may thus be conceived as a well defined subquantum process, in which the field energy is concentrated in a certain region of space in discrete amounts, and the destruction of the particle is just the opposite process. At the quantum-mechanical level the precise details of this process are not significant, and therefore can be ignored. This in fact is done in the current quantum theory, which discusses the creation and destruction of particles as merely a kind of popping in and out of existence with special creation and destruction operators in the mathematics. However, with very fast high-energy processes the results may well depend on these subquantum-mechanical details. And if this should be the case, then the current quantum theory would not be adequate for the treatment of such processes.
The original analogy used by Bohm for developing the idea of the subquantum field is a postulated similarity with the electromagnetic field. The analogy appears in his 1952 articles in Physical Review, and reappears often in later works. The subquantum field exerts a force on the particle in a way that is analogous to the way that the electromagnetic field exerts a force on a charge. And just as the electromagnetic field obeys Maxwell’s equations, so too the subquantum field obeys Schrödinger’s equation. In both cases a complete specification of the field at a given instant over every point in space determines the values of the fields for all times. And in both cases once the physicist knows the field fluxions, he can calculate the force on a particle, so that if he also knows the initial position and momentum of the particle, he can calculate its entire trajectory. Physicists are not yet able to make experiments that localize the position and momentum to a region smaller than that in which the intensity of the hidden subquantum field is applicable. Therefore Bohm notes that they cannot yet find clear-cut experimental evidence that the hypothesis of the hidden variables is necessary.
There are also noted dissimilarities from the electromagnetic wave (or negative analogies as Hesse would say). These dissimilarities are the distinctive aspects of the quantum world in contrast to the classical world. One noteworthy dissimilarity is that the Schrödinger equation is homogeneous while Maxwell’s equations are inhomogeneous, with the result that unlike the electromagnetic field, the subquantum field is not radiated or absorbed, but simply changes its form while its intensity remains constant. In his later works Bohm says that the quantum wave does not impart energy to the particle, but instead functions as a guidance condition, while the particle moves with its own energy.
This feature gives rise to Bohm’s concept of active information, which he introduces in his Undivided Universe. He describes the concept of active information by an analogy with a radio wave, which guides a ship propelled by its own much greater energy while piloted under the guidance of the radio signal. Analogously the elementary particle moves by its own energy under the guidance of the quantum wave. The quantum wave does not push or pull the particle, but rather guides it like the radio wave guides the ship. Bohm explains the two-slit interference experiment in terms of his concept of active information. If both slits are open, the quantum wave passes through both slits while the particle passes through only one slit, and the quantum wave contains information about the slits. As the particle reaches certain points in front of the slits, it is informed to accelerate or decelerate accordingly. Bohm says that the electron particle with its own energy source may have a complex and subtle inner structure, perhaps comparable to a radio receiver.
Quite notably Bohm says that the action of the quantum potential upon the particle depends only on its form and not on its magnitude, and that this implies the possibility of a strong nonlocal connection of distant particles and a strong dependence of the particle on its general environmental context. The forces between particles depend on the wave function of the whole system, so that there is what Bohm calls indivisible wholeness, reminiscent of the organic wholeness of a living being in which the very nature of each part depends on the whole. This absence of the mutual externality and separability of all elements which is characteristic of the classical world makes the quantum world very elusive to the grasp by the physicists’ instruments. But Bohm says it is real and is more basic than the classical world. According to Bohm’s theory the classical world’s autonomy emerges wherever the quantum potential is so relatively small that it can be neglected. But the classical world is actually an abstraction from the subtle quantum world, which is the ultimate ground for existence. These considerations lead to Bohm’s thesis of the implicate order, the order of the quantum world, which supersedes the Cartesian order of the classical world and its mathematics.
In several of his publications Bohm uses the analogy of the lens and the hologram to illustrate the implicate order in ordinary experience. The classical world is like a lens, which produces an approximate correspondence of points on an object to points on an image. In contrast the quantum world is like a hologram, in which each region of the hologram makes possible an image of the whole object. The hologram does not look like the represented object at all, but rather the image is implicit or enfolded. Bohm adds that the term enfolded is not merely a metaphor, but is to be taken literally, and that the order in the hologram is implicate. He also says that there are algebras of the implicate order, and he exemplifies some in his Undivided Universe.
Bohm’s most noteworthy analogy is given in the third chapter of Undivided Universe, where he develops the basic principles of his ontological interpretation in the context of the one-body system. He begins with what is known as the “WKB approximation” for the classical limit in quantum mechanics, and concludes with an equation of motion containing separate terms for both classical and quantum forces, and describing the electron as a particle that has a well defined position that varies continuously, is causally determined, and is never separated from a new type of quantum field that affects the particle. F. David Peat, Bohm’s editor, explains Bohm’s development of this analogy in his biography’s seventh chapter titled “Hidden Variables”. Bohm later rejected Einstein’s idea that the probabilistic results of quantum theory are the result of underlying deterministic motions of smaller particles, as in the Brownian motion analogy. Bohm knew that something analogous to the quantum theory’s wave-particle ambiguity already existed in classical physics. In the nineteenth century the Irish mathematician W.R. Hamilton had shown that it is mathematically possible to recast Newton’s laws about the movement of particles into a description involving waves.
Bohm also knew that Hamilton’s approach is used in quantum theory as an approximation which simplifies calculations, the WKB approximation, which Peat says has a position midway between classical and quantum mechanics with its assumption that quantum particles move along actual trajectories. Unlike most physicists, Bohm took the WKB approximation realistically instead of instrumentally, i.e., as merely a convenient approximation. Peat reports that Bohm’s strategy was to ask what would have to be added to Hamilton’s approach in order to transform this mathematical approximation technique into an equation that can reproduce all the results of quantum theory exactly, and that Bohm’s answer was to introduce his radically new quantum potential, in order to explain all the nonclassical effects. Peat reports that Bohm thus dispensed with metaphysical ideas like Heisenbergian potentialities and actualities, collapsing wave functions, and irreducible probabilities.
In his Beyond Measure: Modern Physics, Philosophy, and the Meaning of Quantum Theory James E. Baggott states that today the Broglie-Bohm theory retains a small, dedicated following within the community of concerned physicists and philosophers, but remains firmly outside the mainstream of quantum physics and appears in few textbooks on the subject. But at the expense of some additional complexity the Broglie-Bohm theory yields the same empirical results as the majority view, thus exemplifying the pragmatist thesis of scientific pluralism.
Bohm’s Critique of Heisenberg’s Copenhagen Interpretation
Shortly after the publication of Heisenberg’s Physics and Philosophy: The Revolution in Modern Science (1958) Bohm wrote an article in The British Journal for the Philosophy of Science (February, 1962) titled “Classical and Nonclassical Concepts in the Quantum Theory.” In a footnote Bohm comments that his article was originally planned as a review of Physics and Philosophy, but since he and Heisenberg had on previous occasions criticized one another’s views, Bohm decided to subtitle his article “An Answer to Heisenberg’s Physics and Philosophy.” On the first page of this article Bohm says that since Heisenberg’s book presents the basic features of the Copenhagen interpretation in such a clear light, it constitutes a useful basis on which further criticisms can be developed. And in this paper Bohm sets forth his own criticisms, one ontological and the other semantical. In summary Bohm’s ontological criticism is that in the exposition of his Copenhagen interpretation Heisenberg introduces ideas that are subjectivist and inconsistent. Bohm incorrectly believes that in expounding his doctrine of potentia Heisenberg states that whereas possibilities can exist outside the human mind, physical actuality can only exist when someone perceives it.
Consider firstly Bohm’s ontological critique. This requires examination of Heisenberg’s statements about the rôle of subjectivism in quantum theory. Heisenberg’s version of the Copenhagen interpretation is set forth in the third chapter of his Physics and Philosophy, which is titled “The Copenhagen Interpretation of Quantum Theory.” He begins by comparing experiments in classical and quantum physics. In both types of physical experiments there are measurement errors, which can be described by probability functions. There is error that is not a property of the observed system, but rather is the experimenter’s ignorance of the true (error-free) measurement. Thus Heisenberg recognizes such a subjective interpretation of the probability function describing measurement error.
But Heisenberg states several times in his exposition that in the case of a quantum experiment the probability function combines both objective and subjective elements in the experimental situation, or as he also says, it represents both statements of a fact and statements of our knowledge of the fact. The statement of our incomplete knowledge of the fact is the measurement error, which is subjective, and it may be different for different experimenters, presumably because the different experimenters do not make exactly the same errors when making their individual measurements. And he comments that the subjective element in the probability function may be practically negligible, and the physicist can then speak of a pure case.
But the statement of fact is a statement about objective possibilities or tendencies, and he references Aristotle’s concept of potentia. The potentia or potential is completely objective and does not depend on any observer. The realization of the transition from the possible to the actual takes place during the act of observation, when the object interacts with the measuring device. Heisenberg explicitly issues a caveat, namely that this transition applies to the physical and not to the psychical act of observation, that it is not connected with the act of registration of the result by the mind of the observer, and that quantum theory does not contain genuinely subjective features, because it does not introduce the mind of the physicist as a part of the atomic event. These comments suggest that Heisenberg wishes to preclude any metaphysical idealism such as Berkeley’s esse est percipi.
But Bohm argues that this caveat is inconsistent with Heisenberg’s other statements about subjectivism. Thus while denying that the transition from the possible to the actual in the measurement operation is connected with the act of registration of the measurement result by the mind of the observer, Heisenberg states that the discontinuous change in the probability function due to the second measurement takes place with the act of registration, because it is a discontinuous change of our knowledge in the instant of registration, a change that has its image in the discontinuous change in the probability function. Bohm quotes this passage in his article, and he concludes that until an observer actually perceives the result of observation, so that he can write a new wave function representing the actual state to which the previous possibilities have collapsed as a result of his perception, there is no actuality at all as far as anything that can appear in the theory is concerned, but only the set of possibilities.
Bohm illustrates his view of Heisenberg’s subjectivism with a hypothetical experiment involving a set of Geiger counters arranged in a grid and toward which a free electron is directed. He supposes a point in time at which the electron has already entered the grid system and has triggered off one of the counters, and furthermore supposes that no observer has yet looked to see which counter has been triggered. Bohm alleges that on Heisenberg’s view, at the supposed point in time when no observer has yet seen which counter has been triggered, one knows the objective possibilities, namely that the counter in question must be one of those located where the amplitude of the electron wave function is appreciable; but if one tries to describe the physical actuality of which counter has been triggered, there is no way in the theory to do so, because the probability function describes only psychic actualities. In other words until an observer actually perceives which counter has operated, so that he can write a new wave function representing the actual state to which the previous possibilities have collapsed as a result of his perception, there is no actuality described by the theory but only the set of possibilities.
Thus the physical actualities play no part at all in the theory, because no predicted result would be changed if the theory were developed without mentioning the physical actualities. It is noteworthy that Bohm seems not merely to be saying that the Copenhagen interpretation is subjectivist because it is probabilistic, and he is not merely criticizing Heisenberg’s Copenhagen interpretation by assuming the subjectivist interpretation of probability as the only probability interpretation. Bohm’s criticism is the claim that the subjectivism is in Heisenberg’s Copenhagen interpretation, and that Heisenberg’s argument unintentionally but logically implies that esse est percipi by the registering mind of the observer.
But such is not Heisenberg’s view. Heisenberg’s thesis is that to be is to be produced by the disturbing physical apparatus used by the observer. Thus Bohm’s thought experiment involving the grid of Geiger counters demonstrates only that there may be a time interval between production and perception of the new actuality. The confusion seems to arise from Heisenberg’s decision to give the quantum probability function both a subjective and an objective interpretation. Normally these two interpretations are distinguished as alternatives, and for good reason: On the objective interpretation the probability function is a statement in real supposition in the object language like any other theory in physics with a semantics describing the real physical world. Thus the probability function so understood is an object language statement with a semantics describing the potentia ontology and the ontology of indeterminism.
On the subjective interpretation the probability function is a statement in logical supposition in a metalanguage for physics with a semantics describing the physicist’s state of knowledge or ignorance expressed by the object language, and it consists of statements making statistical estimates of measurement error. Heisenberg combines these two interpretations, merely because in quantum theory it is not possible to write a separate probability function just for the dispersion due to measurement error. In classical physics, which traditionally assumes an ontology of determinism, variations in repeated measurements under the same experimental conditions are assumed to be due entirely to randomly distributed measurement errors. These could be represented statistically by the standard deviation about the calculated mean of the measurement values, also known as the standard error of the estimate of the true measurement value. In quantum theory, on the other hand, the indeterminist ontology introduces a random variation not originating in measurement errors even though the measurement process captures the indeterminacy. Therefore, the two sources of variation are inseparable, and the effects of both are taken up in the probability function of quantum theory.
What is noteworthy is that Heisenberg does not state in his exposition that the discontinuity in the probability function is due to subjective measurement errors occurring in the second measurement, but rather says that it is a discontinuous change in our knowledge in the instant of registration that has its image in the discontinuous change of the probability function, and is occasioned by the disturbance due to measurement action. Thus the objective interpretation would seem to be the operative one in this passage, because the probability function is viewed as constituting the experimenter’s knowledge in real supposition in object language rather than describing that knowledge in metalanguage. The knowledge that Heisenberg says is the image of the new probability function is the semantics that describes the new physical actuality realized by the action of the measurement apparatus.
However, Bohm prefers to construe this passage to mean that the probability function should be taken with a subjective interpretation, and that it describes knowledge, which Heisenberg calls psychical instead of physical. This places the quantum theory entirely in logical supposition in the metalanguage for physics. Thus Bohm says that the physical actualities play no part whatsoever in the theory, since no predicted result would be changed in any way at all, if the theory were developed without mentioning them. Then Bohm says that to avoid subjectivism, Heisenberg adopts the completely metaphysical assumption of physical entities, which play no part in the theory, but which are introduced to avoid what would otherwise be an untenable philosophical position.
Having exposed to his satisfaction the inconsistency in the Copenhagen interpretation, namely the subjectivism he believes implied in Heisenberg’s exposition and contradicted by Heisenberg’s ad hoc attempt to introduce physical actuality, Bohm then goes on to say that it was due to this problem that he himself was led to criticize the Copenhagen interpretation years earlier, and that while trying to find a way to remedy the absence of the actuality function, he developed his own alternative interpretation. In his alternative interpretation, namely the hidden-variable interpretation, he proposes in addition to the Schrödinger wave function the existence of a particle having a well defined position and momentum, which interacts with the wave in a certain prescribed manner. The position of this particle plays the part of an actuality function in the sense that, when the wave function spreads out over many possibilities, this particle function determines which of these possibilities is actually present. Such is Bohm’s ontological criticism of Heisenberg’s Copenhagen interpretation.
Consider secondly Bohm’s semantical critique. Bohm’s semantical critique is distinctive, because it exploits Heisenberg’s distinction between on the one hand everyday concepts and on the other hand the Newtonian concepts of classical physics that are said to be refinements of the everyday concepts. Bohm rejects the Copenhagen thesis that the classical concepts are necessary for describing macrophysical objects such as the equipment used in microphysical experiments, and thus maintains that alternatives to the Copenhagen interpretation are conceptually possible. He maintains contrary to Heisenberg that the everyday concepts actually used in ordinary experience including the physicist’s description of his laboratory equipment may be refined to “topological” concepts, and need not be the Cartesian-coordinate concepts of Newtonian physics. He therefore believes that these topological concepts are more fundamental in the mathematical sense for the description of space and time than the Cartesian concepts, and that the latter must be translated onto the former.
Bohm exemplifies this idea with the problem of locating an ordinary pencil. The location is not ordinarily stated in terms of a coordinate system such as latitude and longitude. What is actually done in ordinary experience is to locate the pencil as laying upon a desk within a certain room in a certain house, which is located on a certain street, etc. Thus the pencil is located with the aid of a series of topological relations, in which one entity is within or upon another entity. He then says that the laboratory physicist also uses topological relations in his work. In no experiment does he ever locate anything by giving an exact coordinate, which is to say an infinite number of decimals. Rather what he does in practice for making a measurement is to place a pointer between certain marks on a scale, thus locating it by the topological relation of between the marks. In every experiment the notion of precisely defined coordinates is just an abstraction, which is approximated when a topologically described experimental result is translated into the Cartesian language of continuous coordinates. Bohm adds that everyday concepts could be refined in other ways than to topological concepts, but that for description of space and time, the topological concepts are most appropriate for physical theory.
Bohm then suggests a topological formulation of the quantum theory, and says that these nonclassical concepts make possible new kinds of experimental predictions, which cannot be considered in the framework of the Copenhagen interpretation, and which according to Heisenberg’s conclusions are not possible. Bohm says that there is a remarkable analogy between the mathematics of topology and that of the modern quantum mechanical field theory, and that utilization of this analogy can make possible the development of a topological formulation, which while leading to the results of the usual quantum theory in suitable limiting cases, nevertheless possesses certain genuinely novel features with regard both to its mathematical formalism and to its experimental predictions. He also says that he cannot go into the details in this paper, and he is not known to have done so in any other paper he has published.
This novation with or without the inspiring analogy is a promissory note backed by an as-yet-unearned income, because Bohm does not set forth an explicit topological formulation of the quantum theory. If he actually had set forth such a new formulation, and if its novel experimental predictions were found to be superior to those made by the current quantum theory, such as resolving the renormalization problem, then his new topological formulation would be a revolutionary development in microphysical theory, and much more than merely another interpretation of the quantum theory. The relevance of a topological description of macroscopic instruments is that they are vague. The significance of Heisenberg’s “everyday” concepts is merely that they are vague enough that they do not make either Newtonian or quantum claims by signifying any microphysical realities.